We propose and theoretically and numerically investigate narrowband integrated filters consisting of identical resonant dielectric ridges on the surface of a single-mode dielectric slab waveguide. The proposed composite structures operate near a bound state in the continuum (BIC) and enable spectral filtering of transverse-electric-polarized guided modes propagating in the waveguide. We demonstrate that by proper choice of the distances between the ridges, flat-top reflectance profiles with steep slopes and virtually no sidelobes can be obtained using just a few ridges. In particular, the structure consisting of two ridges can optically implement the second-order Butterworth filter, whereas at a larger number of ridges, excellent approximations to higher-order Butterworth filters can be achieved. Owing to the BIC supported by the ridges constituting the composite structure, the flat-top reflection band can be made arbitrarily narrow without increasing structure size. In addition to the filtering properties, the investigated structures support another type of BIC—the Fabry–Perot BIC—arising when the distances between adjacent ridges meet the Fabry–Perot resonance condition. In the vicinity of the Fabry–Perot BIC, an effect similar to electromagnetically induced transparency is observed, namely, sharp transmittance peaks against the background of a wide transmittance dip.